Question

$n$ drops of water, each of radius $2mm$, fall through air at a terminal velocity of $8cm/s$. If they coalesce to form a single drop, then the terminal velocity of the combined drop is $32cm/s$. The value of $n$ is

Answer 1

$\frac{4}{3}\pi R^3=n\times \frac{4}{3}\pi r^3$
or $R^3=n{r}^3$ or $R=n^{1/3}r$
or $R=2{\pi }^{1/3}mm$
$v_0\propto r^2,v_0^{\prime }\propto R^2$
Now, $\frac{v_0^{\prime }}{v_0}=\frac{R^2}{r^2}=\frac{4n^{2/3}}{4}f$
or $\frac{32}{8}=n^{2/3}$ or $n^{2/3}=4$
or $n=4^{3/2}=\sqrt{64}$
or $n=8$.